We are early 20s in age, and we work together. We have mutual friends on facebook.

I think she is attracted to me because of the following:

-I picked something up in front of her (something light and I am a muscular guy) and she joked that I couldnt handle it.
-She laughs when I'm around
-When I asked her name, she flirted with me and covered her name tag and said "I dont have one..."

-recently i pulled on her sweatshirt and scared her, and later in the night I squeezed by her and someone else in a tight space very lightly and she yelled to me "watch where your walking!"

But she doesnt know I know her name. She never told it to me and I never used it since i see her sparingly.

I work with her in over a week. Should I wait or facebook her? Does it give away my intentions? We have mutual work friends on facebook

If you're comfortable with building attraction through facebook chat, then go for it, but it's quite different to doing so in person, it's more psychological than physical, obviously.

If otherwise, continue with in person until you've built a little more attraction.

thing is, if I am the "friender" it shows I have "done my research" on the internet looking for her which looks pathetic no?

keep in mind I have a girl, this girl is just hot as hell

If you have a legitimate reason for coming across her and you want to converse with her, then go for it. If you feel that you are not ready for the potential idea that others are aware that you wanted to add this girl, then the answer should be obvious.